Name:Nora Country:Canada State:British Columbia Birthday:1/1/1986 Gender:Female
Interests:Music, math, all the fun stuff. :-) Expertise:Scaring Gr 8's and 9's, sleeping on the bus, etc. Occupation:Student Industry:Education/Research
Unit Plan Math 8 - Solving Equations (MathPower 8 Chapter 6) For a unit of 10 75-min classes.
Rationale/Overview: Algebra is important to the study of mathematics because it simplifies and clarifies situations with symbolic representations and allows people to study more complicated problems by simplifying and clarifying their thinking process. It is similar to the use of pronouns in English where they help condense expressions. In this unit, students will be exposed, possibly for the first time, to the principles of working with algebraic equations (or statements). Algebra also allows people to manipulate mathematical objects without knowing what they are in specific. This allows for generalization, which applies to all fields of mathematics. They will be learning how to solve simple algebraic equations and how to use these equations to solve simple word problems.
PLO's: B2 model and solve problems using linear equations of the form ax = b x/a = b, a ≠ 0 ax + b = c x/a+ b = c , a ≠ 0 a(x + b) = c concretely, pictorially, and symbolically, where a, b, and c are integers
Background Teacher Preparations: - Should have done the fraction and integer units - Should have done the variable expressions, patterns, and relations unit
Cross-Curricular Connections: - Simple reading comprehension - Presentation skills - Research and teamwork skills
Extensions and Adaptations: - Will depend on the specific student who needs the extensions or adaptations.
Resources: - complicated passages - balance puzzles - secret codes and coding equations - summary of classic algebra-friendly word problems
Overview of lessons: Lesson 1. - PLOs: none - Objective: Introduction to algebraic equations 6.1: Translating word problems into algebraic equations - Teaching Strategies: group work, summarizing group work with lecture - Student Activities: In groups, they will translate English passages into symbols. (Different groups can get the same passage. They can compare their representations afterwards.) - Assessment Strategies: Each group submits their symbolic representation and a legend with their given passage. They will be marked on clarity and accuracy. - Materials: 3-6 English passages that can be translated into symbols
Lesson 2. - PLOs: none - Objectives: 6.2+6.3: SWBAT solve one-step equations by guessing or by addition/subtraction - Teaching Strategies: lecture (for examples), student volunteers for balance puzzles - Student Activities: volunteer for balance puzzles - Assessment Strategies: Check if students are following by finding volunteers to do the balance puzzles - Materials: Balance puzzles with one-step addition or subtraction and their corresponding equations (a good 20 of them)
Lesson 3. - PLOs: B2 i and ii - Objectives: 6.4+6.5: SWBAT solve one-step equations by multiplication or division - Teaching Strategies: Same as Lesson 2 - Student Activities: Same as Lesson 2 - Assessment Strategies: Same as Lesson 2 - Materials: Balance puzzles with one-step multiplication or division and their corresponding equations (a good 20 of them)
Lesson 4. - PLOs: B2 iii to v - Objectives: 6.6-6.8: SWBAT solve multi-step linear equations with mixed operations - Teaching Strategies: Same as Lesson 2 - Student Activities: Same as Lesson 2 - Assessment Strategies: Same as Lesson 2 - Materials: Balance puzzles with multi-step mixed operations and their corresponding equations (~10 of them)
Lesson 5 ("I'm thinking of a number"). - PLOs: B2 iii to v - Objectives: 6.8: Same as Lesson 4 - Teaching Strategies: lecture (example), pair work - Student Activities: Students will each think of a secret number, then do a series of operations to that number and tell their partner the result. The parter will work out the secret number by undoing the operations. Repeat 3-5 times with increasing number of operations. (could change partners in between.) - Assessment Strategies: Have students show their work for the activity on a sheet of paper. Hand in at the end of class for completion mark and formative assessment. - Materials: none
Lesson 6. - PLOs: B2 i to v - Objectives: Review 6.2-6.8 - Teaching Strategies: group work - Student Activities: Students will form 5 to 6 groups. Each group will be given a line of secret code to decode using a linear equation. After the decoding, the class will play a game of simplified wheel of fortune to guess each other's secret code and coding equation. - Assessment Strategies: Have each group write down their work in decoding their secret code and hand in at the end of class for completion mark and formative assessment. - Materials: 6 lines of secret codes and their coding equations
Lesson 7. - PLOs: none - Objectives: 6.9: SWBAT solve word problems using algebraic equations - Teaching Strategies: group work - Student Activities: Students will be put into groups. Each group will be given a classic "hard" (algebra-friendly) word problem. They will first be asked to come up with a solution without using algebra, then come up with a "standard" algebraic solution. Have each group present their two solutions. Groups can exchange problems and repeat the process if time allows. As a preparation for Lesson 9, students will be asked to form groups of 3 or 4 for the history research project. - Assessment Strategies: Have each group submit a good copy of their two solutions to at least 1 problem for formative assessment and preparation for next class. - Materials: a collection of classic word problems
Lesson 8. - PLOs: none - Objectives: 6.9: SWBAT identify connections between algebraic solutions and other solutions - Teaching Strategies: lecture (show examples), group work (same groups as last class) - Student Activities: Students will go through the two solutions to a word problem generated by another group last class and identify corresponding steps that are the same in the two different representations. Groups will present their findings. As preparation for the next class, groups formed last class for the history research project will be asked to submit an area of research that they will be presenting. - Assessment Strategies: Have each group write up a list of their findings and hand in at the end of the class. - Materials: Solution write-ups from last class.
Lesson 9. - PLOs: none - Objectives: to promote students' interest in mathematics by its historical connections - Teaching Strategies: research - Student Activities: Students will research on their submitted topic in groups in the computer lab. They will be asked to find at least 5 interesting facts in the area of their research to present to the class. If time allows, presentations can start during this lesson. - Assessment Strategies: Peer evaluation on the presentation by project rubric. Each group will also hand in a written copy of their list of facts to me for reference. - Materials: book the computer lab
Lesson 10. - PLOs: none - Objectives: Same as Lesson 9 - Teaching Strategies: presentation, review session for the test if time allows - Student Activities: Students will present what they found in their research. - Assessment Strategies: Peer evaluation on the presentation by project rubric. - Materials: review questions for the test
Project details: History research project
Teaching/learning objective: - to promote students' interest in mathematics by its historical connections Content: In groups of 3 or 4, students will research on a topic of their choice that's relevant to algebra and history and find 5 interesting facts in their area of research. Each group will present these facts with any background information that they think are necessary or interesting. Each group will also hand in a written copy of these facts and their background information to me for my reference.
Marking: This project is completely peer-marked. Project rubric: (each on a scale of 5) - Informative ("I didn't know this!" and "I was just wondering about that!") - Interesting (as opposed to "Who cares about that...") - Presentation ("Easy to follow." and "Loud and clear.") - (Group mate only) Cooperation ("Easy to work with and did his/her share of the work.") Space for feedback should be provided on the peer evaluation form as well.
Rationale: The purpose of this project is to promote student interest. It is a small project with little work and a lot of freedom of choice because of that. It is purely peer-marked because of the purpose as well, since if the peers thought it was interesting, then the objective has been met.
A possible question on the unit test: Write down one interesting fact that you have learned from another group's presentation for the history research project.
Bridge: This lesson will give a different perspective on solving algebraic equations.
Teaching objectives: - to help students be aware of alternative representations of ideas they learn in class. - to promote interactions among students while learning.
Student objectives: SWBAT - B2 model and solve problems using linear equations of the form ax + b = c x/a+ b = c , a ≠ 0 a(x + b) = c concretely and symbolically, where a, b, and c are integers
Pretest: If following the unit plan, students should be able to solve one-step linear equations. Students should also have been exposed to solving multi-step linear equations with mixed operations in the previous class.
Participatory Activity: - Start the class with routine timed free writing. Topic: equation (can be changed according to what I want to know about the students on that day.) -------- 5-7mins - Short review of the idea that an equation is like a balance, and methods of solving multi-step linear equations using this idea -------- ~5mins - Introduce the idea that an equation is a statement. Use 2x+5=19 as an example. Show the method of solving the equation as "undoing" operations to the "secret number" x. -------- ~5mins - Give a more complicated example, like (3(x+2)+6)/5=9. Use the analogy of unwrapping presents. -------- ~5mins - Have a student volunteer give another equation and have a different student volunteer solve it. -------- 1-2mins - Have students form pairs and have each student make up an equation of two operations for their partner to solve. Have them show their work on a sheet of paper per student to be handed in at the end of class. -------- ~5-10mins - Have students switch partners and each make up an equation of 3 operations for their partner to solve. Show work on the same sheet of paper. -------- ~5-10mins - Have students switch partners again and each make up an equation of 4 operations for their partner to solve. Show work on the same sheet of paper. -------- ~ 5-10mins - As an alternative, introduce the idea that an equal sign is like a mirror. Solving equations can be thought of as isolating the variable by moving other things across the "mirror", which changes any operation into its opposite operation. Again, use 2x+5=19 as an example. -------- ~5mins - Give an example where x is on both sides. For example 2x-3=x. Solve it using the mirror idea. -------- ~5mins - Homework -------- ~10mins or until the end of class.
Post-test: If most students can solve their partners' equations correctly, the student objectives are met.
Lesson 6: Decoding and Wheel of Fortune (modified)
Bridge: Today we will play a game to review the techniques for solving linear algebraic equations that we have learned in the last two weeks.
Teacher objective: - to have students enjoy math while learning, to cure math-phobia. - to promote interactions among students while learning.
Student objectives: SWBAT - B2 model and solve problems using linear equations of the form ax = b x/a = b, a ≠ 0 ax + b = c x/a+ b = c , a ≠ 0 a(x + b) = c concretely and symbolically, where a, b, and c are integers
Pretest: If following the unit plan, the class should have learned to solve multi-step linear equations with mixed operations.
Participatory Activity: - Start the class with routine timed free writing. Topic: algebra (can be changed according to what I want to know about the students on that day.) -------- 5-7mins - Students will form groups of 5 or 6 -------- 1-2mins - Students will decode the messages that are 20 to 30 characters long each -------- ~10mins - Each group will take turn going up to host a game of Wheel of Fortune (with no scoring) where the rest of the class guess for letters that are in the message until someone can reveal the message. The class will also "guess" for their coding equation at the same time. -------- ~10mins per group, so ~50mins together. - If there is time left over, clean up and do homework.
Post-test: If most students are able to decode the messages and participate in the game, the student objectives are met.
Conclusion: Next class we will be looking at the use of algebra in word problems.
Bridge: Algebra is not an isolated topic. It can be used to clarify and simplify situations for us in many places. Word problems is one of these places.
Teacher objectives: - to partially cure the fear of unfamiliar problems - to show that math is interconnected. - to show that there is possibly more than one solution to a problem.
Student objectives: SWBAT - translate word problems into algebraic equations and solve them using algebra.
Pretest: If following the unit plan, students should be able to solve multi-step linear equations with mixed operations.
Participatory Activity: - Start the class with routine timed free writing. Topic: problems (can be changed according to what I want to know about the students on that day.) -------- 5-7mins - Show a classic "hard" (algebra-friendly) word problem and give the students some time to look for a solution by thinking on their own or talking to a neighbour. -------- ~5mins - If there are people who can do it, have them present their solution (algebraic or not is both ok). If not present a non-algebraic solution. -------- ~5mins - If there are people who can do it, have them present a different solution. Look for an algebraic solution if the first one is non-algebraic, and vise verse. If not present a solution of the type I need. -------- ~5mins - Compare the two solutions and point out similarities and possible corresponding steps. -------- ~5mins - Have students form groups of 5 or 6. -------- 1-2mins - Give each group a classic "hard" algebra-friendly word problem. Have them work out an algebraic and a non-algebraic solution and write up a good copy of both solutions. -------- ~15mins - Have each group present their problem and two solutions. -------- ~5mins per group, so 25mins in total. If run out of time, some presentations can be pushed to next class. - Clean up and form groups of 3 or 4 for the upcoming history research project. Announce the history research project, and have the groups select an area of research by next class. -------- ~5mins
Post-test: If all groups of students successfully came up with an algebraic solution, the learning objectives are met. The non-algebraic solution can be a challenge.
Conclusion: Next class, we will continue on this activity.
-Show how a golden rectangle is constructed and show how this is used to draw spirals
-An example of a building that has been contructed using design based on the golden rectangle
-An example of spirals occuring in nature (nautilus shell)
Part 2: Evaluation of Project
Some of the possible benefits or uses for this project:
-It provides students with a way to connect the math they learn in class with real life images that they see around them.
-It gives students a chance to practice their analyzing skills by connecting key points in the image into recognizable shapes.
-It fosters an appreciation for and awareness of patterns in everyday life
-It encourages the habit of looking for patterns in everyday life.
-It can be used as a cure to geometry-phobia
Some possible weaknesses for this project:
-Students might be tempted to go for simple designes in their project.
-It is hard to set a guideline for the level of complexity of the design used in the project.
-Students might be tempted to go for the more famous or well-known and well-analyzed images.For example, the numerous famous graphic examples of the golden ratio.
Some possible modefications for this project:
-Instead of asking students to find their own images, assign them to pre-selected images.
-Instead of asking students to find images of any sort, limit the images to a particular theme.
-Find two pictures that have similar design framework structures.
-The finished product of this project might be presentations or gallery displays of the images and the design frameworks.
Part 3: New Project Idea
This is our project idea based on our evaluation of the Geometry of Design project:
Project title: Real Sequences
Grade level: 8-10
Purpose(s):
-To draw connections between math and real life
-To give students a chance to practice analyzing skills
-To foster an appreciation for and awareness of patterns in real life, and to encourage the habit of looking for patterns in real life
-To build teamworking skills
Description of activities:
Students will be put into small groups, and each group will pick (or be assigned) a particular type of sequences.For example, arithmatic, geometric, fibonacci, perfect squares, triangular numbers, harmonic.
Similar to Geometry of Design, the groups of students will find occurences of their assigned type of sequences in "real life" (either in nature or in artificial designs) and report back to the class on what they have found.
Length of time: This will be a short project.Our estimate is that 1.5-2 weeks should be enough.In class time should be given for groups to work on the presentation or final product together towards the end of the 2 weeks.
Final product: As the final product of this project, students can be asked to produce posters on the definition of their given type of sequences and the examples they have found in real life for their type of sequences.This can be followed by a group presentation.Another alternative is to have students make posters of the occurences of the type sequences that they have found without revealing the type of sequence, and have the rest of the class figure out the type of sequences found in that example.
Handout:The handout for this project would be a copy of what’s written here with a complete list of types of sequences to choose from.
Marking criteria: Students will be marked on:
By the teacher:
-accuracy of the mathematical concepts
-clarity of presentation (both on the poster and orally)
-quantity of examples found (At least 3 for full marks)
I've taken two pictures of this in case there are parts where it's hard to read on one. Please download and enlarge if you're having trouble reading it at this size.
I really liked this reading.Many of the things the book talked about as part of the “problem solving procedure” are what we take for granted and implement without much thought.However, an explicit list of steps to follow is maybe really what students need when they haven’t developed this procedure from their own experience.I think that this would help reduce the number of people who give up on or develop a fear for problem solving.At least they would know that it’s normal to not get the answer right away, and they wouldn’t be discouraged because of that.On the other hand though, as our professor for the Problem Solving class said in the beginning of the term, what students do in their math classes these days are not “problems”.Those can’t be “problems” when you expect them to do 40 or so in an hour, which is what it’s like on provincial exams.With those questions, it’s pointless to implement the problem solving procedure because it won’t help them much in terms of success in answering the questions.That’s because these questions mostly test the students’ ability to classify questions and memorize a series of steps to apply to each type of question.There’s not much thinking needed compared to what we call “problems”.So before we go into trying to get students to implement this procedure, it might be worthwhile to bring actual problems and the act of problem solving back into math classes.The problem with that is that good problems are hard to find.Most of the problems deemed to be “suitable for Gr 8-10” seem to be the ones that can be easily classified into types, and then students can memorize the steps to solve them.That seems to be how “problem solving” is taught too, by types and by methods to solve them, when this is totally the opposite of what problem solving means to me.Sometimes I feel like I can’t get out of the classification trap unless I go far enough up in difficulty, but by the time I find one, it’s already too hard for the vast majority of the students I’ll be teaching.
Stories of things that surprised me and things I've learned during the short practicum:
First thing that surprised me... well, maybe not really. I half expected it... is that math teachers are boring. They really are surprisingly rigid and uninspiring. Each class is mostly overhead notes and examples of a certain textbook section, and then half an hour of homework. I feel the pressure when I go into that environment that I need to somehow continue this "tradition" because if I did something different, I'd be risking not being able to drive the routines into the students' heads, the routines that are outlined in the textbook or the worksheets or curriculum or exams... I wonder if this is really the only way math can be taught. On the other hand, I do understand that there were people who tried, and they eventually gave up because students are conditioned to accept this kind of math classes, and consider an ideal math class in my mind, where people get to solve exciting problems or do more "meaningful" things with math, just beyond them... Too hard. They don't even want to try. I dropped in and taught one lesson on ratios and proportions to the Gr 8's, and as part of an activity, I asked both classes the question "Do you like math?", and out of a class of about 30, which both classes were, about 3-5 people answered yes, another 3-5 answered maybe, and the rest said no. I knew this was out there, that most people hated math ever since they were little, that math was just something that they have to get over with, but I didn't think it was this serious until I was out there seeing this. I really wanted to ask more about why they didn't like math... Maybe it's only because it's boring, and that would be easy to fix... but maybe it's because of years of conditioning in elementary school that math is hard, and "normal people" aren't supposed to get math... That would be much harder to fix...
